Optimal. Leaf size=923 \[ -\frac{6 b \text{PolyLog}\left (3,-i e^{i (c+d x)}\right ) f^3}{\left (a^2-b^2\right ) d^4}+\frac{6 b \text{PolyLog}\left (3,i e^{i (c+d x)}\right ) f^3}{\left (a^2-b^2\right ) d^4}+\frac{3 a \text{PolyLog}\left (3,-e^{2 i (c+d x)}\right ) f^3}{2 \left (a^2-b^2\right ) d^4}-\frac{6 b^2 \text{PolyLog}\left (4,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right ) f^3}{\left (a^2-b^2\right )^{3/2} d^4}+\frac{6 b^2 \text{PolyLog}\left (4,\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) f^3}{\left (a^2-b^2\right )^{3/2} d^4}+\frac{6 i b (e+f x) \text{PolyLog}\left (2,-i e^{i (c+d x)}\right ) f^2}{\left (a^2-b^2\right ) d^3}-\frac{6 i b (e+f x) \text{PolyLog}\left (2,i e^{i (c+d x)}\right ) f^2}{\left (a^2-b^2\right ) d^3}-\frac{3 i a (e+f x) \text{PolyLog}\left (2,-e^{2 i (c+d x)}\right ) f^2}{\left (a^2-b^2\right ) d^3}+\frac{6 i b^2 (e+f x) \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right ) f^2}{\left (a^2-b^2\right )^{3/2} d^3}-\frac{6 i b^2 (e+f x) \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) f^2}{\left (a^2-b^2\right )^{3/2} d^3}-\frac{6 i b (e+f x)^2 \tan ^{-1}\left (e^{i (c+d x)}\right ) f}{\left (a^2-b^2\right ) d^2}+\frac{3 a (e+f x)^2 \log \left (1+e^{2 i (c+d x)}\right ) f}{\left (a^2-b^2\right ) d^2}+\frac{3 b^2 (e+f x)^2 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right ) f}{\left (a^2-b^2\right )^{3/2} d^2}-\frac{3 b^2 (e+f x)^2 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) f}{\left (a^2-b^2\right )^{3/2} d^2}-\frac{i a (e+f x)^3}{\left (a^2-b^2\right ) d}+\frac{i b^2 (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d}-\frac{i b^2 (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d}-\frac{b (e+f x)^3 \sec (c+d x)}{\left (a^2-b^2\right ) d}+\frac{a (e+f x)^3 \tan (c+d x)}{\left (a^2-b^2\right ) d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.93669, antiderivative size = 923, normalized size of antiderivative = 1., number of steps used = 29, number of rules used = 13, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.464, Rules used = {4533, 3323, 2264, 2190, 2531, 6609, 2282, 6589, 6742, 4184, 3719, 4409, 4181} \[ -\frac{6 b \text{PolyLog}\left (3,-i e^{i (c+d x)}\right ) f^3}{\left (a^2-b^2\right ) d^4}+\frac{6 b \text{PolyLog}\left (3,i e^{i (c+d x)}\right ) f^3}{\left (a^2-b^2\right ) d^4}+\frac{3 a \text{PolyLog}\left (3,-e^{2 i (c+d x)}\right ) f^3}{2 \left (a^2-b^2\right ) d^4}-\frac{6 b^2 \text{PolyLog}\left (4,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right ) f^3}{\left (a^2-b^2\right )^{3/2} d^4}+\frac{6 b^2 \text{PolyLog}\left (4,\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) f^3}{\left (a^2-b^2\right )^{3/2} d^4}+\frac{6 i b (e+f x) \text{PolyLog}\left (2,-i e^{i (c+d x)}\right ) f^2}{\left (a^2-b^2\right ) d^3}-\frac{6 i b (e+f x) \text{PolyLog}\left (2,i e^{i (c+d x)}\right ) f^2}{\left (a^2-b^2\right ) d^3}-\frac{3 i a (e+f x) \text{PolyLog}\left (2,-e^{2 i (c+d x)}\right ) f^2}{\left (a^2-b^2\right ) d^3}+\frac{6 i b^2 (e+f x) \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right ) f^2}{\left (a^2-b^2\right )^{3/2} d^3}-\frac{6 i b^2 (e+f x) \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) f^2}{\left (a^2-b^2\right )^{3/2} d^3}-\frac{6 i b (e+f x)^2 \tan ^{-1}\left (e^{i (c+d x)}\right ) f}{\left (a^2-b^2\right ) d^2}+\frac{3 a (e+f x)^2 \log \left (1+e^{2 i (c+d x)}\right ) f}{\left (a^2-b^2\right ) d^2}+\frac{3 b^2 (e+f x)^2 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right ) f}{\left (a^2-b^2\right )^{3/2} d^2}-\frac{3 b^2 (e+f x)^2 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) f}{\left (a^2-b^2\right )^{3/2} d^2}-\frac{i a (e+f x)^3}{\left (a^2-b^2\right ) d}+\frac{i b^2 (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d}-\frac{i b^2 (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d}-\frac{b (e+f x)^3 \sec (c+d x)}{\left (a^2-b^2\right ) d}+\frac{a (e+f x)^3 \tan (c+d x)}{\left (a^2-b^2\right ) d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4533
Rule 3323
Rule 2264
Rule 2190
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rule 6742
Rule 4184
Rule 3719
Rule 4409
Rule 4181
Rubi steps
\begin{align*} \int \frac{(e+f x)^3 \sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac{\int (e+f x)^3 \sec ^2(c+d x) (a-b \sin (c+d x)) \, dx}{a^2-b^2}-\frac{b^2 \int \frac{(e+f x)^3}{a+b \sin (c+d x)} \, dx}{a^2-b^2}\\ &=\frac{\int \left (a (e+f x)^3 \sec ^2(c+d x)-b (e+f x)^3 \sec (c+d x) \tan (c+d x)\right ) \, dx}{a^2-b^2}-\frac{\left (2 b^2\right ) \int \frac{e^{i (c+d x)} (e+f x)^3}{i b+2 a e^{i (c+d x)}-i b e^{2 i (c+d x)}} \, dx}{a^2-b^2}\\ &=\frac{\left (2 i b^3\right ) \int \frac{e^{i (c+d x)} (e+f x)^3}{2 a-2 \sqrt{a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{\left (a^2-b^2\right )^{3/2}}-\frac{\left (2 i b^3\right ) \int \frac{e^{i (c+d x)} (e+f x)^3}{2 a+2 \sqrt{a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{\left (a^2-b^2\right )^{3/2}}+\frac{a \int (e+f x)^3 \sec ^2(c+d x) \, dx}{a^2-b^2}-\frac{b \int (e+f x)^3 \sec (c+d x) \tan (c+d x) \, dx}{a^2-b^2}\\ &=\frac{i b^2 (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d}-\frac{i b^2 (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d}-\frac{b (e+f x)^3 \sec (c+d x)}{\left (a^2-b^2\right ) d}+\frac{a (e+f x)^3 \tan (c+d x)}{\left (a^2-b^2\right ) d}-\frac{\left (3 i b^2 f\right ) \int (e+f x)^2 \log \left (1-\frac{2 i b e^{i (c+d x)}}{2 a-2 \sqrt{a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right )^{3/2} d}+\frac{\left (3 i b^2 f\right ) \int (e+f x)^2 \log \left (1-\frac{2 i b e^{i (c+d x)}}{2 a+2 \sqrt{a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right )^{3/2} d}-\frac{(3 a f) \int (e+f x)^2 \tan (c+d x) \, dx}{\left (a^2-b^2\right ) d}+\frac{(3 b f) \int (e+f x)^2 \sec (c+d x) \, dx}{\left (a^2-b^2\right ) d}\\ &=-\frac{i a (e+f x)^3}{\left (a^2-b^2\right ) d}-\frac{6 i b f (e+f x)^2 \tan ^{-1}\left (e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}+\frac{i b^2 (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d}-\frac{i b^2 (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d}+\frac{3 b^2 f (e+f x)^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^2}-\frac{3 b^2 f (e+f x)^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^2}-\frac{b (e+f x)^3 \sec (c+d x)}{\left (a^2-b^2\right ) d}+\frac{a (e+f x)^3 \tan (c+d x)}{\left (a^2-b^2\right ) d}+\frac{(6 i a f) \int \frac{e^{2 i (c+d x)} (e+f x)^2}{1+e^{2 i (c+d x)}} \, dx}{\left (a^2-b^2\right ) d}-\frac{\left (6 b^2 f^2\right ) \int (e+f x) \text{Li}_2\left (\frac{2 i b e^{i (c+d x)}}{2 a-2 \sqrt{a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right )^{3/2} d^2}+\frac{\left (6 b^2 f^2\right ) \int (e+f x) \text{Li}_2\left (\frac{2 i b e^{i (c+d x)}}{2 a+2 \sqrt{a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right )^{3/2} d^2}-\frac{\left (6 b f^2\right ) \int (e+f x) \log \left (1-i e^{i (c+d x)}\right ) \, dx}{\left (a^2-b^2\right ) d^2}+\frac{\left (6 b f^2\right ) \int (e+f x) \log \left (1+i e^{i (c+d x)}\right ) \, dx}{\left (a^2-b^2\right ) d^2}\\ &=-\frac{i a (e+f x)^3}{\left (a^2-b^2\right ) d}-\frac{6 i b f (e+f x)^2 \tan ^{-1}\left (e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}+\frac{i b^2 (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d}-\frac{i b^2 (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d}+\frac{3 a f (e+f x)^2 \log \left (1+e^{2 i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}+\frac{6 i b f^2 (e+f x) \text{Li}_2\left (-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}-\frac{6 i b f^2 (e+f x) \text{Li}_2\left (i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}+\frac{3 b^2 f (e+f x)^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^2}-\frac{3 b^2 f (e+f x)^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^2}+\frac{6 i b^2 f^2 (e+f x) \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^3}-\frac{6 i b^2 f^2 (e+f x) \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^3}-\frac{b (e+f x)^3 \sec (c+d x)}{\left (a^2-b^2\right ) d}+\frac{a (e+f x)^3 \tan (c+d x)}{\left (a^2-b^2\right ) d}-\frac{\left (6 a f^2\right ) \int (e+f x) \log \left (1+e^{2 i (c+d x)}\right ) \, dx}{\left (a^2-b^2\right ) d^2}-\frac{\left (6 i b^2 f^3\right ) \int \text{Li}_3\left (\frac{2 i b e^{i (c+d x)}}{2 a-2 \sqrt{a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right )^{3/2} d^3}+\frac{\left (6 i b^2 f^3\right ) \int \text{Li}_3\left (\frac{2 i b e^{i (c+d x)}}{2 a+2 \sqrt{a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right )^{3/2} d^3}-\frac{\left (6 i b f^3\right ) \int \text{Li}_2\left (-i e^{i (c+d x)}\right ) \, dx}{\left (a^2-b^2\right ) d^3}+\frac{\left (6 i b f^3\right ) \int \text{Li}_2\left (i e^{i (c+d x)}\right ) \, dx}{\left (a^2-b^2\right ) d^3}\\ &=-\frac{i a (e+f x)^3}{\left (a^2-b^2\right ) d}-\frac{6 i b f (e+f x)^2 \tan ^{-1}\left (e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}+\frac{i b^2 (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d}-\frac{i b^2 (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d}+\frac{3 a f (e+f x)^2 \log \left (1+e^{2 i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}+\frac{6 i b f^2 (e+f x) \text{Li}_2\left (-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}-\frac{6 i b f^2 (e+f x) \text{Li}_2\left (i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}+\frac{3 b^2 f (e+f x)^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^2}-\frac{3 b^2 f (e+f x)^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^2}-\frac{3 i a f^2 (e+f x) \text{Li}_2\left (-e^{2 i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}+\frac{6 i b^2 f^2 (e+f x) \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^3}-\frac{6 i b^2 f^2 (e+f x) \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^3}-\frac{b (e+f x)^3 \sec (c+d x)}{\left (a^2-b^2\right ) d}+\frac{a (e+f x)^3 \tan (c+d x)}{\left (a^2-b^2\right ) d}-\frac{\left (6 b^2 f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (\frac{i b x}{a-\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{\left (a^2-b^2\right )^{3/2} d^4}+\frac{\left (6 b^2 f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (\frac{i b x}{a+\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{\left (a^2-b^2\right )^{3/2} d^4}-\frac{\left (6 b f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^4}+\frac{\left (6 b f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^4}+\frac{\left (3 i a f^3\right ) \int \text{Li}_2\left (-e^{2 i (c+d x)}\right ) \, dx}{\left (a^2-b^2\right ) d^3}\\ &=-\frac{i a (e+f x)^3}{\left (a^2-b^2\right ) d}-\frac{6 i b f (e+f x)^2 \tan ^{-1}\left (e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}+\frac{i b^2 (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d}-\frac{i b^2 (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d}+\frac{3 a f (e+f x)^2 \log \left (1+e^{2 i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}+\frac{6 i b f^2 (e+f x) \text{Li}_2\left (-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}-\frac{6 i b f^2 (e+f x) \text{Li}_2\left (i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}+\frac{3 b^2 f (e+f x)^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^2}-\frac{3 b^2 f (e+f x)^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^2}-\frac{3 i a f^2 (e+f x) \text{Li}_2\left (-e^{2 i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}-\frac{6 b f^3 \text{Li}_3\left (-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^4}+\frac{6 b f^3 \text{Li}_3\left (i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^4}+\frac{6 i b^2 f^2 (e+f x) \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^3}-\frac{6 i b^2 f^2 (e+f x) \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^3}-\frac{6 b^2 f^3 \text{Li}_4\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^4}+\frac{6 b^2 f^3 \text{Li}_4\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^4}-\frac{b (e+f x)^3 \sec (c+d x)}{\left (a^2-b^2\right ) d}+\frac{a (e+f x)^3 \tan (c+d x)}{\left (a^2-b^2\right ) d}+\frac{\left (3 a f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{2 i (c+d x)}\right )}{2 \left (a^2-b^2\right ) d^4}\\ &=-\frac{i a (e+f x)^3}{\left (a^2-b^2\right ) d}-\frac{6 i b f (e+f x)^2 \tan ^{-1}\left (e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}+\frac{i b^2 (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d}-\frac{i b^2 (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d}+\frac{3 a f (e+f x)^2 \log \left (1+e^{2 i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}+\frac{6 i b f^2 (e+f x) \text{Li}_2\left (-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}-\frac{6 i b f^2 (e+f x) \text{Li}_2\left (i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}+\frac{3 b^2 f (e+f x)^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^2}-\frac{3 b^2 f (e+f x)^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^2}-\frac{3 i a f^2 (e+f x) \text{Li}_2\left (-e^{2 i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}-\frac{6 b f^3 \text{Li}_3\left (-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^4}+\frac{6 b f^3 \text{Li}_3\left (i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^4}+\frac{6 i b^2 f^2 (e+f x) \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^3}-\frac{6 i b^2 f^2 (e+f x) \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^3}+\frac{3 a f^3 \text{Li}_3\left (-e^{2 i (c+d x)}\right )}{2 \left (a^2-b^2\right ) d^4}-\frac{6 b^2 f^3 \text{Li}_4\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^4}+\frac{6 b^2 f^3 \text{Li}_4\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^4}-\frac{b (e+f x)^3 \sec (c+d x)}{\left (a^2-b^2\right ) d}+\frac{a (e+f x)^3 \tan (c+d x)}{\left (a^2-b^2\right ) d}\\ \end{align*}
Mathematica [A] time = 9.37484, size = 1438, normalized size = 1.56 \[ \frac{b \sec (c) (e+f x)^3}{\left (b^2-a^2\right ) d}+\frac{f \left (\frac{2 i a (e+f x)^3}{f}+\frac{3 (a-b) \left (1+e^{2 i c}\right ) \log \left (1-i e^{-i (c+d x)}\right ) (e+f x)^2}{d}+\frac{3 (a+b) \left (1+e^{2 i c}\right ) \log \left (1+i e^{-i (c+d x)}\right ) (e+f x)^2}{d}+\frac{6 (a+b) \left (1+e^{2 i c}\right ) f \left (i d (e+f x) \text{PolyLog}\left (2,-i e^{-i (c+d x)}\right )+f \text{PolyLog}\left (3,-i e^{-i (c+d x)}\right )\right )}{d^3}+\frac{6 (a-b) \left (1+e^{2 i c}\right ) f \left (i d (e+f x) \text{PolyLog}\left (2,i e^{-i (c+d x)}\right )+f \text{PolyLog}\left (3,i e^{-i (c+d x)}\right )\right )}{d^3}\right )}{\left (a^2-b^2\right ) d \left (1+e^{2 i c}\right )}+\frac{b^2 \left (2 \sqrt{b^2-a^2} e^3 \tan ^{-1}\left (\frac{i a+b e^{i (c+d x)}}{\sqrt{a^2-b^2}}\right ) d^3+\sqrt{a^2-b^2} f^3 x^3 \log \left (1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right ) d^3+3 \sqrt{a^2-b^2} e f^2 x^2 \log \left (1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right ) d^3+3 \sqrt{a^2-b^2} e^2 f x \log \left (1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right ) d^3-\sqrt{a^2-b^2} f^3 x^3 \log \left (\frac{e^{i (c+d x)} b}{i a+\sqrt{b^2-a^2}}+1\right ) d^3-3 \sqrt{a^2-b^2} e f^2 x^2 \log \left (\frac{e^{i (c+d x)} b}{i a+\sqrt{b^2-a^2}}+1\right ) d^3-3 \sqrt{a^2-b^2} e^2 f x \log \left (\frac{e^{i (c+d x)} b}{i a+\sqrt{b^2-a^2}}+1\right ) d^3-3 i \sqrt{a^2-b^2} f (e+f x)^2 \text{PolyLog}\left (2,\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right ) d^2+3 i \sqrt{a^2-b^2} f (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right ) d^2+6 \sqrt{a^2-b^2} e f^2 \text{PolyLog}\left (3,\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right ) d+6 \sqrt{a^2-b^2} f^3 x \text{PolyLog}\left (3,\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right ) d-6 \sqrt{a^2-b^2} e f^2 \text{PolyLog}\left (3,-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right ) d-6 \sqrt{a^2-b^2} f^3 x \text{PolyLog}\left (3,-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right ) d+6 i \sqrt{a^2-b^2} f^3 \text{PolyLog}\left (4,\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right )-6 i \sqrt{a^2-b^2} f^3 \text{PolyLog}\left (4,-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right )\right )}{\sqrt{-\left (a^2-b^2\right )^2} \left (b^2-a^2\right ) d^4}+\frac{\sin \left (\frac{d x}{2}\right ) e^3+3 f x \sin \left (\frac{d x}{2}\right ) e^2+3 f^2 x^2 \sin \left (\frac{d x}{2}\right ) e+f^3 x^3 \sin \left (\frac{d x}{2}\right )}{(a+b) d \left (\cos \left (\frac{c}{2}\right )-\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )-\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}+\frac{\sin \left (\frac{d x}{2}\right ) e^3+3 f x \sin \left (\frac{d x}{2}\right ) e^2+3 f^2 x^2 \sin \left (\frac{d x}{2}\right ) e+f^3 x^3 \sin \left (\frac{d x}{2}\right )}{(a-b) d \left (\cos \left (\frac{c}{2}\right )+\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )+\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 2.507, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{3} \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{a+b\sin \left ( dx+c \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 7.73611, size = 9469, normalized size = 10.26 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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